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Your friend is wrong. When you cast it from your graveyard, it has no memory at all about what it targeted previously, and the self-replacement effect (gets 10 counters instead) only means that instead of adding 5 counters to whatever you're targeting, you add 10 instead. This doesn't modify the 5 counters that were placed a long time ago when it resolved the first time.

You'll get a total of 15 counters if you cast it twice. 5 the first time (on any creature), 10 the second time (on any creature). 

Here's how it works:

1.  You announce that you're casting Increasing Savagery .  The spell goes on the stack.
2.  You choose a creature to target.
3.  You determine the total casting cost for the card.  Assuming that nothing is altering the cast of the card, you either pay if you are casting it from your hand, or if you are casting it from your graveyard.
4.  As the spell resolves, the target gets counters.  If the spell was cast from your hand, the target gets five counters.  If the spell was cast from your graveyard, the target gets ten counters.
5.  If the spell was cast from your hand, it goes to the graveyard.  If it was cast from the graveyard, exile it.

Here's where your confusion is coming from:  If you cast the spell from your hand and you target Runeclaw Bear , you do not have to target that same creature when you cast it from your graveyard.  You're overthinking how the card works.  Instead of thinking of Increasing Savagery as one spell, think of it as two different ones.  If you have two Increasing Savagery s in your hand, would you automatically think that the second one would have to target the same creature the first one did?  No, you wouldn't.  If you think of casting it from your graveyard as casting a brand new spell (which you technically are doing anyway), hopefully you'll be in the same mindset as you would be when casting a seperate card from your hand.